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Some fixed point generalizations are not real generalizations. (English) Zbl 1251.54045

The authors show how several recent results in metric fixed point theory which appear as generalizations of previous results, can in fact readily be deduced as corollaries of the latter.

This concerns mainly contractive conditions, where some expression such as $d\left(fx,fy\right)\le k\phantom{\rule{0.166667em}{0ex}}d\left(x,y\right)$ is replaced by something like $d\left(fx,fy\right)\le k\phantom{\rule{0.166667em}{0ex}}d\left(gx,gy\right)$. While the latter condition is more general, one can return to the first case by considering a map $h$ defined on the range of $g$ by $h\left(gx\right)=fx$.

The authors apply this transformation to derive a host of results concerning contractive mappings in cone metric spaces, multivalued contractions in complete metric spaces, and nonexpansive mappings between metric spaces and Banach spaces.

Reviewer’s remarks: For cone metric spaces, some of the results mentioned in the paper under review can be reduced even further to the well-known situation in metric spaces, cf. S. Janković, Z. Kadelburg and S. Radenović [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 7, 2591–2601 (2011; Zbl 1221.54059)]. In this regard, note also that Y.-Q. Feng and W. Mao [Fixed Point Theory 11, No. 2, 259–264 (2010; Zbl 1221.54055)] proved that, for a cone metric space $X$ with cone metric $d$ and cone $P$ taking values in a Banach space, $D\left(x,y\right)=inf\left\{\parallel u\parallel :u\in P,\phantom{\rule{4pt}{0ex}}u\ge d\left(x,y\right)\right\}$ defines a metric on $X$ so that $\left(X,D\right)$ is complete if and only if $\left(X,d\right)$ is complete.

By taking into account such reductions, authors and referees should in principle be able to improve the overall significance of research papers in this field. Unfortunately, it appears unlikely that the increase of superfluous publications can be reversed as easily as their insignificance can be exposed.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces